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Introduction to Probability, | 2nd Edition | ISBN: 9781886529236 | Authors: Dimitri P. Bertsekas John N. Tsitsiklis
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Consider a game show with an infinite pool of contestants,

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QUESTION:

Consider a game show with an infinite pool of contestants, where at each round i, contestant i obtains a number by spinning a continuously calibrated wheel. The contestant with the smallest number thus far survives. Successive wheel spins are independent and we assume that there are no ties. Let N be the round at which contestant 1 is eliminated. For any positive integer n, find peN = n).

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QUESTION:

Consider a game show with an infinite pool of contestants, where at each round i, contestant i obtains a number by spinning a continuously calibrated wheel. The contestant with the smallest number thus far survives. Successive wheel spins are independent and we assume that there are no ties. Let N be the round at which contestant 1 is eliminated. For any positive integer n, find peN = n).

ANSWER:

Step 1 of 2

You have that the required probability

         

Here for gives you the event when a contestant i number is the smallest number obtained of the number of contestants starting from 1 up to j.

You also have

         

Now you can claim that the probability subject to the condition 

         

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